These questions are taken from the 2015 (April and October) IFoA papers, with additional parts
in some cases to give them a flavour of the UEA exams
The marks available are based on the IFoA papers being 100 marks for a 3 hour paper.
From April 2015 Q2
2) The table below shows cumulative claim amounts incurred on a portfolio of insurance
policies.
Accident Year
Development Year
0
1
2
3
2011
1,509
1,969
2,106
2,207
2012
1,542
2,186
2,985
2013
1,734
1,924
2014
1,773
Annual premiums written in 2014 were 4,013 and the ultimate loss ratio has been
estimated as 93.5%. Claims can be assumed to be fully run off by the end of
development year 3.
i) Estimate the total claims arising from policies written in 2014 only, using the
Bornhuetter-Ferguson method. [7]
ii) Discuss the advantages and disadvantages of using the BF method compared to other methods
you could use [3]
iii) What other issues should the actuary consider when deciding the reserves [3]
[Total 13]
From April 2015 Q3
3) (i) (a) Explain why an insurance company might purchase reinsurance.
(b) Describe two types of reinsurance. [3]
The claim amounts on a particular type of insurance policy follow a Pareto distribution with
mean 270 and standard deviation 340.
(ii) Determine the lowest retention amount such that under excess of loss reinsurance the
probability of a claim involving the reinsurer is 5%. [4]
[Total 7]
From April 2015 Q8
8) The number of claims, N, in a given year on a particular type of insurance policy is
given by:
P(N = n) = 0.8  0.2 n n = 0, 1, 2, …
Individual claim amounts are independent from claim to claim and follow a Pareto
distribution with parameters   5 and   1,000 .
(i) Calculate the mean and variance of the aggregate annual claims per policy. [4]
(ii) Calculate the probability that aggregate annual claims exceed 400 using:
(a) a Normal approximation.
(b) a Lognormal approximation.
[6]
(iii) Explain which approximation in part (ii) you believe is more reliable. [2]
[Total 12]
From April 2015 Q9
9) Let p be an unknown parameter and let f ( p | x) be the probability density of the posterior
distribution of p given information x .
(i) Explain why under all-or-nothing loss the Bayes estimate of p is the mode of f ( p | x) [2]
John is setting up an insurance company to insure luxury yachts. In year 1 he will insure 100
yachts and in year 2 he will insure 100 + g yachts where g is an integer.
If there is a claim the insurance company pays a fixed sum of $1m per claim.
The probability of a claim on a policy in a given year is p. You may assume that the probability
of more than one claim on a policy in any given year is zero. Prior beliefs about p are described
by a Beta distribution with parameters   2 and   8 .
In year 1 total claims are $13m and in year 2 they are $20m.
(ii) Derive the posterior distribution of p in terms of g. [4]
(iii) Show that it is not possible in this case for the Bayes estimate of p to be the same under
quadratic loss and all-or-nothing loss. [6]
[Total 12]
From October 2015 Q1
5) An actuary is simulating claims Xi on a portfolio of insurance policies.
For each i, let Yi be 1 if Xi exceeds a given amount M and 0 if not. The variance of Yi is 0.12.
The actuary wishes to estimate the proportion of claims that exceed M.
Calculate the number of simulations that the actuary will have to perform in order to estimate the true
proportion to within 0.01 with 99% confidence. [3 marks]
From October 2015 Q4
4) A small island is holding a vote on independence. Two recent survey results are
shown below:
Poll
A
B
Sample size
10
20
Support for independence
5
11
You should assume that the samples are independent.
A politician is using a suitable uniform distribution as the prior distribution in order
to estimate the proportion  in favour of independence.
(i) Calculate an estimate of  under the quadratic loss function.
[3]
A rival politician decides to use instead a beta distribution as the prior, with
parameters  and  , where    .
(ii) Determine the new estimate of  under the “all-or-nothing” loss function in
terms of  .
[Total 7]
[4]
From October 2015 Q5
5 Claims X each year from a portfolio of insurance policies are normally distributed with mean
 and variance  2 . Prior information is that  is normally distributed with known mean  and
known variance  2 .
Aggregate claims over the last n years have been xi for i = 1 to n, and you should assume that
these are independent.
(i) Derive the posterior distribution of  .
[5]
(ii) Write down the Bayesian estimate of  under quadratic loss.
[1]
(iii) Show that the estimate in your answer to part (ii) can be expressed in the form of a
credibility estimate, including statement of the credibility factor Z.
[2]
[Total 8]
From October 2015 Q8 (extended)
8 The run-off triangle below shows cumulative claims incurred on a portfolio of general
insurance policies
Development Year
Policy Year
0
1
2
3
2011
1,528
2,034
2,212
2,310
2012
1,812
2,251
2,951
2013
1,693
1,851
2014
2,125
Annual premiums written in 2014 were 4,023 and the ultimate loss ratio has been estimated as
91%. Claims paid to date for policy year 2014 are 572.
i) Estimate the outstanding claims to be paid arising from policies written in 2014 only, using the
Bornheutter-Ferguson technique, stating any assumptions that you make.
[9]
ii) Re-calculate the BF reserves on the assumption that you had not been given a loss ratio and had to
estimate this from the data.
iii) Calculate the total reserves using the chain ladder method
[6]
iv) If the inflation rates (from mid year to mid year0 have been
2011-2012: 12%
2012-2013: 15%
2013-2014: 8%
and the future expected level of inflation from mid 2014 onwards is 5%, calculate (without
discounting) the inflation adjusted reserves.
[9]
v) If the number of claims paid in each year have been:
Development Year
Policy Year
0
1
2
2011
14
7
2
2012
16
3
5
2013
12
2
2014
18
Use the average cost per claim method to calculate:
a) How may outstanding claims you expect there to be in total
b) What the total reserves should be
[Total 33]
3
1
[9]
October 2015 Q9 (hard perhaps beyond syllabus)
9 A random variable X follows a gamma distribution with parameters  and  .
i) Derive the moment generating function (MGF) of X. [3]
ii) Derive the coefficient of skewness of X. [8]
iii) Explain why you think skewness matters for general insurance actuaries [3]
iv) In what circumstances might an actuary consider using a gamma distribution [3]
[Total 17]
April 2011 Q3
3 Let y1 ,…, y n be samples from a uniform distribution on the interval [0, θ] where θ > 0 is an
unknown constant. Prior beliefs about θ are given by a distribution with density
   (1 )
 
f ( )  
0
otherwise

where α and β are positive constants.
(i) Show that the posterior distribution of θ given y1 is of the same form as the prior distribution,
specifying the parameters involved. [4]
(ii) Write down the posterior distribution of θ given y1 ,…, y n [2]
[Total 6]
April 2011 Q4
4 The annual number of claims on an insurance policy within a certain portfolio follows a
Poisson distribution with mean μ. The parameter μ varies from policy to policy and can be
considered as a random variable that follows an exponential distribution with mean 1

Find the unconditional distribution of the annual number of claims on a randomly chosen policy from
the portfolio. [6]
April 2011 Q10
10 The number of claims on a portfolio of insurance policies has a Poisson distribution with
mean 200. Individual claim amounts are exponentially distributed with mean 40.
The insurance company calculates premiums using a premium loading of 40% and is considering
entering into one of the following re-insurance arrangements:
(A) No reinsurance.
(B) Individual excess of loss insurance with retention 60 with a reinsurance company that
calculates premiums using a premium loading of 55%.
(C) Proportional reinsurance with retention 75% with a reinsurance company that calculates
premiums using a premium loading of 45%.
(i) Find the insurance company’s expected profit under each arrangement. [6]
(ii) Find the probability that the insurer makes a profit of less than 2000 under each of the
arrangements using a normal approximation. [8]
[Total 14]
September 2011 Q2
2 An accountant is using a psychic octopus to predict the outcome of tosses of a fair coin. He
claims that the octopus has a probability p > 0.5 of successfully predicting the outcome of any
given coin toss.
His actuarial colleague is extremely skeptical and summarises his prior beliefs about p as
follows: there is an 80% chance that p = 0.5 and a 20% chance that p is uniformly distributed on
the interval [0.5,1].
The octopus successfully predicts the results of 7 out of 8 coin tosses.
Calculate the posterior probability that p = 0.5.
September 2011 Q3
Loss amounts under a class of insurance policies follow an exponential distribution with mean 100.
The insurance company wishes to enter into an individual excess of loss reinsurance arrangement
with retention level M set such that 8 out of 10 claims will not involve the reinsurer.
(i) Find the retention M. [2]
For a given claim, let X I denote the amount paid by the insurer and X R the amount
paid by the reinsurer.
(ii) Calculate E( X I ) and E( X R ) [3]
[Total 5]
September 2011 Q4
4 Claims on a portfolio of insurance policies follow a Poisson process with parameter λ.
The insurance company calculates premiums using a premium loading of θ and has an initial
surplus of U.
(i) Define the surplus process U(t). [1]
(ii) Define the probabilities ψ(U, t) and ψ(U). [2]
(iii) Explain how ψ(U, t) and ψ(U) depend on λ. [2]
[Total 5]
September 2011 Q7
7 A portfolio of insurance policies contains two types of risk. Type I risks make up 80% of claims and
give rise to loss amounts which follow a normal distribution with mean 100 and variance 400. Type II
risks give rise to loss amounts which are normally distributed with mean 115 and variance 900.
(i) Calculate the mean and variance of the loss amount for a randomly chosen claim. [3]
(ii) Explain whether the loss amount for a randomly chosen claim follows a normal distribution.
[2]
The insurance company has in place an excess of loss reinsurance arrangement with retention
130.
(iii) Calculate the probability that a randomly chosen claim from the portfolio results in a
payment by the reinsurer. [3]
(iv) Calculate the proportion of claims involving the reinsurer that arise from Type II risks. [2]
[Total 10]